{"title":"Overdetermined elliptic problems in nontrivial contractible domains of the sphere","authors":"David Ruiz , Pieralberto Sicbaldi , Jing Wu","doi":"10.1016/j.matpur.2023.10.009","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we prove the existence of nontrivial contractible domains <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, such that the overdetermined elliptic problem<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mi>ε</mi><msub><mrow><mi>Δ</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>u</mi><mo>+</mo><mi>u</mi><mo>−</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mn>0</mn></mtd><mtd><mtext>in Ω, </mtext></mtd></mtr><mtr><mtd><mi>u</mi><mo>></mo><mn>0</mn></mtd><mtd><mtext>in Ω, </mtext></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn></mtd><mtd><mtext>on ∂Ω, </mtext></mtd></mtr><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>ν</mi></mrow></msub><mi>u</mi><mo>=</mo><mtext>constant</mtext></mtd><mtd><mtext>on ∂Ω, </mtext></mtd></mtr></mtable></mrow></math></span></span></span> admits a positive solution. Here <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> is the Laplace-Beltrami operator in the unit sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with respect to the canonical round metric <em>g</em>, <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span> is a small real parameter and <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mfrac><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span> (<span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span> if <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>). These domains are perturbations of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>∖</mo><mi>D</mi></math></span>, where <em>D</em><span><span> is a small geodesic ball. This shows in particular that Serrin's theorem for </span>overdetermined problems<span> in the Euclidean space cannot be generalized to the sphere even for contractible domains.</span></span></p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"180 ","pages":"Pages 151-187"},"PeriodicalIF":2.1000,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782423001502","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
In this paper, we prove the existence of nontrivial contractible domains , , such that the overdetermined elliptic problem admits a positive solution. Here is the Laplace-Beltrami operator in the unit sphere with respect to the canonical round metric g, is a small real parameter and ( if ). These domains are perturbations of , where D is a small geodesic ball. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains.
期刊介绍:
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.